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An accelerated primal-dual flow for linearly constrained multiobjective optimization

Hao Luo, Qiaoyuan Shu, Xinmin Yang

TL;DR

This work extends continuous-time primal-dual dynamics to linearly constrained multiobjective optimization by introducing an accelerated multiobjective primal-dual flow (AMPD) and a novel merit function Pi that links feasibility to weak Pareto optimality. Through Lyapunov analysis, the authors establish exponential decay in the continuous model and derive an IMEX discretization (AMPD-QP) that yields provable rates: O(1/k) for feasibility and O(1/k^2) for the objective gap under convex and strongly convex scenarios. The AMPD-QP scheme reduces to a quadratic subproblem at each step and demonstrates strong empirical performance, including favorable Pareto front distributions in high-dimensional tests. Overall, the paper provides a principled bridge between continuous-time accelerated dynamics and discrete multiobjective optimization, with practical implications for efficiently solving linearly constrained multiobjective problems.

Abstract

In this paper, we propose a continuous-time primal-dual approach for linearly constrained multiobjective optimization problems. A novel dynamical model, called accelerated multiobjective primal-dual flow, is presented with a second-order equation for the primal variable and a first-order equation for the dual variable. It can be viewed as an extension of the accelerated primal-dual flow by Luo [arXiv:2109.12604, 2021] for the single objective case. To facilitate the convergence rate analysis, we introduce a new merit function, which motivates the use of the feasibility violation and the objective gap to measure the weakly Pareto optimality. By using a proper Lyapunov function, we establish the exponential decay rate in the continuous level. After that, we consider an implicit-explicit scheme, which yields an accelerated multiobjective primal-dual method with a quadratic subproblem, and prove the sublinear rates of the feasibility violation and the objective gap, under the convex case and the strongly convex case, respectively. Numerical results are provided to demonstrate the performance of the proposed method.

An accelerated primal-dual flow for linearly constrained multiobjective optimization

TL;DR

This work extends continuous-time primal-dual dynamics to linearly constrained multiobjective optimization by introducing an accelerated multiobjective primal-dual flow (AMPD) and a novel merit function Pi that links feasibility to weak Pareto optimality. Through Lyapunov analysis, the authors establish exponential decay in the continuous model and derive an IMEX discretization (AMPD-QP) that yields provable rates: O(1/k) for feasibility and O(1/k^2) for the objective gap under convex and strongly convex scenarios. The AMPD-QP scheme reduces to a quadratic subproblem at each step and demonstrates strong empirical performance, including favorable Pareto front distributions in high-dimensional tests. Overall, the paper provides a principled bridge between continuous-time accelerated dynamics and discrete multiobjective optimization, with practical implications for efficiently solving linearly constrained multiobjective problems.

Abstract

In this paper, we propose a continuous-time primal-dual approach for linearly constrained multiobjective optimization problems. A novel dynamical model, called accelerated multiobjective primal-dual flow, is presented with a second-order equation for the primal variable and a first-order equation for the dual variable. It can be viewed as an extension of the accelerated primal-dual flow by Luo [arXiv:2109.12604, 2021] for the single objective case. To facilitate the convergence rate analysis, we introduce a new merit function, which motivates the use of the feasibility violation and the objective gap to measure the weakly Pareto optimality. By using a proper Lyapunov function, we establish the exponential decay rate in the continuous level. After that, we consider an implicit-explicit scheme, which yields an accelerated multiobjective primal-dual method with a quadratic subproblem, and prove the sublinear rates of the feasibility violation and the objective gap, under the convex case and the strongly convex case, respectively. Numerical results are provided to demonstrate the performance of the proposed method.

Paper Structure

This paper contains 22 sections, 15 theorems, 116 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Single objective optimization problems
  3. Multiobjective optimization problems
  4. Organization
  5. Preliminary
  6. Notation
  7. Pareto optimality
  8. Optimality condition
  9. New merit function
  10. Accelerated Multiobjective Primal-Dual Flow
  11. Continuous model
  12. Time rescaling
  13. Lyapunov analysis
  14. Rate of Convergence
  15. An Accelerated Multiobjective Primal-Dual Method
  16. ...and 7 more sections

Key Result

Lemma 2.1

If $f\in\mathcal{S}_{\mu,L}^{1,1}(\,{\mathbb R}^d)$, then

Figures (4)

  • Figure 1: Numerical results of \ref{['eg:eg1']}. From left to right: the iterate trajectory, the approximate Pareto front at different step, the average residual.
  • Figure 2: Numerical results of \ref{['eg:eg2']}. From left to right: the iterate trajectory, the approximate Pareto front at different step, the average residual.
  • Figure 3: Numerical results of \ref{['eg:eg3']}. From left to right: the iterate trajectory, the approximate Pareto front at different step, the average residual.
  • Figure 4: Approximate Pareto fronts of AMPD-QP and ALAMO for selected problems.

Theorems & Definitions (35)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.1
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 25 more